Inferior Products and Profitable Deception

Inferior Products and Profitable Deception∗
         Paul Heidhues                  Botond Kőszegi                       Takeshi Murooka
            ESMT                  Central European University                University of Munich

                                             December 1, 2014


                                                   Abstract

          We analyze conditions facilitating profitable deception in a simple model of a competitive
      retail market. Firms selling homogenous products set up-front prices that consumers understand
      and additional prices that naive consumers ignore unless revealed to them by a firm, where we
      assume that there is a binding floor on the up-front prices. We show that our reduced-form
      framework captures more elaborate models of the credit-card and mutual-fund markets, and
      argue that it may apply to bank accounts and mortgages as well. Our main results establish
      that “bad” products (those with lower social surplus than an alternative) tend to be more
      reliably profitable than “good” products. Specifically, (1) in a market with a single socially
      valuable product and sufficiently many firms, a deceptive equilibrium—in which firms hide
      additional prices—does not exist and firms make zero profits. But perversely, (2) if the product
      is socially wasteful, then a profitable deceptive equilibrium always exists. Furthermore, (3) in a
      market with multiple products, since a superior product both diverts sophisticated consumers
      and renders an inferior product socially wasteful in comparison, it guarantees that firms can
      profitably sell the inferior product by deceiving consumers.

          JEL Codes: D14, D18, D21




  ∗
    The setup of this paper was developed in our earlier working paper “The Market for Deceptive Products,” which
we split into two papers. The companion paper is titled “Exploitative Innovation.” Kőszegi thanks the Berkeley
Center for Equitable Growth, and Heidhues and Kőszegi thank the European Research Council (Starting Grant
#313341) for financial support. We are grateful to Özlem Bedre-Defolie, Stefano DellaVigna, Drew Fudenberg,
Michael Grajek, Michael Grubb, Péter Kondor, Nicolas Melissas, Antoinette Schoar, and seminar and conference
audiences for useful comments and feedback.


                                                       1
1        Introduction

In this paper, we investigate circumstances under which firms sell products by deceiving some con-
sumers about the products’ full price, focusing (in contrast to much of the literature) on deception
that leads to positive equilibrium profits in seemingly competitive industries.1 We identify a novel,
perverse aspect of profitable deception: products that generate lower social surplus than the best
alternative facilitate deception precisely because they would not survive in the market if consumers
understood their full price, and therefore firms often make profits on exactly such bad products
but not on good products. We argue that—in seeking these profits—firms disproportionately push
bad products and might enter markets for bad products in potentially massive numbers, creating
further economic inefficiencies.
        We develop our insights in the reduced-form model of Section 2.1, which builds on the seminal
theory of Gabaix and Laibson (2006). Firms are engaged in simultaneous-move price competition
to sell a single homogenous product. Each firm charges a transparent up-front price as well as an
additional price, and unless at least one firm (costlessly) unshrouds the additional prices, naive
consumers ignore these prices when making purchase decisions. To capture the notion that for
some products firms cannot return all profits from later charges by lowering initial charges, we
deviate from most existing work and posit that there is a floor on the up-front price.2 We argue
that a profitable deceptive equilibrium—wherein all firms shroud additional prices—is the most
plausible equilibrium whenever it exists: it is then the unique equilibrium in the variant of our
model in which unshrouding carries a cost (no matter how small), and all firms prefer it over an
    1
     Hidden fees have often enabled firms to reap substantial profits despite seemingly considerable competition, at
least at the price-competition stage when entry and marketing costs have been paid and customer bases have been
identified and reached. Investigating trade and portfolio data from a large German bank, for example, Hackethal,
Inderst and Meyer (2010) document that “bank revenues from security transactions amount to e2,560 per customer
per year” (2.4 percent of mean portfolio value), a figure likely well above the marginal cost of serving a customer.
Similarly, based on a number of measures, including the 20-percent average premium in interbank purchases of
outstanding credit-card balances, Ausubel (1991) argues that credit-card companies make large profits. Ellison and
Ellison (2009) describe a variety of obfuscation strategies online computer-parts retailers use, and document that
such strategies can generate surprisingly large profits given the near homogeneity of products. These observations,
however, do not mean that the net economic surplus taking all operating costs into account are large or even positive
in these markets: for example, fixed entry costs can dissipate any profits from the later stage of serving consumers.
   2
      Armstrong and Vickers (2012), Ko (2012), and Grubb (forthcoming) also analyze models with variants of our
price-floor assumption.




                                                         1
unshrouded-prices and therefore zero-profit equilibrium. Hence, in our discussion we presume that
a deceptive equilibrium is played whenever it exists.
   In Section 2.2, we establish that our reduced-form framework captures more elaborate models
of two important markets that have been invoked as conducive to hidden charges: credit cards and
mutual funds. In our model of credit cards, issuers make offers consisting of an up-front price and
an interest rate to naive time-inconsistent borrowers, who derive a convenience benefit from using
a credit card and underestimate how much interest they will pay. In the case of indifference, each
consumer chooses whether to get or charge a card by lexicographically applying an exogenously
given preference over cards. Then, any consumer who prefers a firm’s card will get the card if the
firm charges an up-front price of zero. Any additional consumers a negative up-front price attracts,
therefore, will not use the firm’s card, and hence these consumers are unprofitable. As a result,
firms act as if they were facing a price floor of zero.
   In our model of mutual funds, firms choose front loads investors understand and management
fees investors ignore unless explained to them by a firm. We argue that a key section of the
Investment Company Act of 1940, which prohibits the dilution of existing investors’ shares in
favor of new investors, prevents funds from offering discounts to new consumers, so that the front
load must be non-negative. Finally, we briefly and informally lay out several other applications,
including mortgages with changing terms, bank accounts, printers, and hotels, emphasizing that in
some of these markets there may not be a binding floor on the up-front price.
   Section 3 presents our basic results. As two benchmark cases, we show that if the price floor
is not binding, only a zero-profit deceptive equilibrium exists, and note that if consumers are
sophisticated in that they observe and take into account additional prices, firms again earn zero
profits. If the price floor is binding and consumers are naive, however, profitable deception may
occur. If other firms shroud and the up-front price is at the floor, a firm cannot compete on the
up-front price and can compete on the total price only if it unshrouds—but because consumers
who learn of the additional prices may not buy the product, the firm may find the latter form of
competition unattractive. If this is the case for all firms, a profitable deceptive equilibrium exists.
Otherwise, additional prices are unshrouded with probability one, and firms earn zero profits.



                                                   2
The above condition for a firm to find unshrouding unattractive has some potentially important
implications for when profitable deception occurs. First, if the product is socially wasteful (its value
relative to consumers’ outside option is lower than its production cost), a firm that unshrouds cannot
go on to profitably sell its product, so no firm ever wants to unshroud. Perversely, therefore, in
a socially wasteful industry a profitable deceptive equilibrium always exists. But if the product is
socially valuable, a firm that would make sufficiently low profits from deception can earn higher
profits from unshrouding and capturing the entire market, so if there is such a firm only a non-
deceptive, zero-profit equilibrium exists. Hence, because in an industry with many firms some firm
earns low profits, entry into socially valuable industries makes these industries more transparent;
and whenever deceptive practices survive in an industry with many firms, our model says that
the industry is socially wasteful. We also show that if consumers initially underappreciate the
additional price because they are unaware of a valuable add-on they want to spend on, only an
unshrouded-prices equilibrium exists. From the perspective of our model, therefore, the observation
that unshrouding has not taken place in the credit-card market implies that excessive credit-card
borrowing is not only unanticipated, but unwanted by consumers.
   In Section 4, we extend our model by assuming that there are both sophisticated and naive
consumers in the market. Consistent with the predictions of Gabaix and Laibson (2006) and Arm-
strong and Vickers (2012) that sophisticated consumers facilitate transparency and efficiency, we
find that if the product is socially valuable and there are sufficiently many sophisticated consumers,
only a non-deceptive equilibrium exists. In contrast to received wisdom, however, we also show that
in a multi-product market with a superior and an inferior product, often sophisticated and naive
consumers self-separate into buying the former and the latter product, respectively, and sophis-
ticated consumers exert no pressure to unshroud the inferior product’s additional prices. Worse,
because the superior product renders the inferior product socially wasteful in relative terms, it
guarantees that profitable deception in the market for the inferior product can be maintained. This
observation has a striking implication: all it takes for profitable deception to occur in a competitive
industry is the existence of an inferior product with a shroudable price component and a binding
floor on the up-front price, and firms’ profits derive entirely from selling this inferior product.



                                                   3
In Section 5, we turn to extensions and modifications of our framework. We show that if a
firm has market power in the superior-product market, it may have an incentive to unshroud to
attract naive consumers to itself, especially if—similarly to for instance Vanguard in the mutual-
fund market—it sells mostly the superior product. But if the firm’s market power is limited and
unshrouding is costly, the extent to which the firm educates consumers is also limited.
    In Section 6, we discuss some policy implications of and evidence for our model. Since deception
is likely to be more common and have more adverse welfare consequences when the price floor is
binding, policymakers should concentrate their regulatory efforts on industries with a binding price
floor—of which supranormal profits is a telltale sign. We also note that a policymaker can improve
outcomes for consumers if she can lower additional prices. And although the overarching prediction
of our paper that inferior products facilitate profitable deception and hence should go together with
it is difficult to test directly, some auxiliary predictions of our model are consistent with existing
evidence. In particular, our framework predicts that firms disproportionately push—for instance
through commission-driven intermediaries or persuasive advertising—inferior products and may
enter the market for inferior products in mass numbers, and that the regulation of hidden fees may
not result in increases in other fees, and hence benefits consumers.
    In Section 7, we discuss the behavioral-economics and classical literatures most closely related
to our paper. While a growing theoretical literature investigates how firms exploit naive consumers
by charging hidden or unexpected fees, our paper goes beyond all of the existing work in making the
central prediction that socially wasteful and inferior products tend to be sold more profitably than
better products, and identifying a number of additional implications of this insight. We conclude
in Section 8 by pointing out important further questions raised by our model.


2     Basic Model and Applications

2.1   Setup

N ≥ 2 firms compete for naive consumers who value each firm’s product at v > 0 and are looking to
buy at most one item. Consumers have an outside option with utility u. Firms play a simultaneous-



                                                  4
move game in which they set up-front prices fn and additional prices an ∈ [0, a], and decide whether
to costlessly unshroud the additional prices. A consumer who buys a product must pay both prices
associated with that product—she cannot avoid the additional price. If all firms shroud, consumers
make purchase decisions as if the total price of product n was fn . If at least one firm unshrouds, all
firms’ additional prices become known to all consumers, and consumers make purchase decisions
based on the true total prices fn + an .3 If consumers weakly prefer buying and are indifferent
between a number of firms, each of these firms gets a positive market share, with this share being
sn ∈ (0, 1) if consumers are indifferent between all firms.
       Although we focus most of our analysis and discussion on price misperceptions, we also allow
for consumers to misperceive the product’s value, as when investors overestimate the gross return
of a managed mutual fund. Paralleling our setup for additional prices, we assume that if all firms
shroud, consumers perceive the value of the product to be ṽ, and if at least one firm unshrouds,
consumers correctly understand the value to be v.
       Firm n’s cost of providing the product is cn > 0. We let cmin = minn {cn }, and—to ensure that
our industry is competitive in the corresponding classical Bertrand model—assume that cn = cmin
for at least two firms n. In addition, we assume that max{v, ṽ + a} > cn for all n; a firm with
max{v, ṽ + a} < cn cannot profitably sell its product, so we think of it as not participating in the
market. And to avoid uninteresting qualifications, we suppose that v − u 6= cmin .
       Deviating from much of the literature, we impose a floor on the up-front price: fn ≥ f . We
assume that f ≤ min{ṽ, v} − u, which means that whether or not unshrouding occurs, consumers
are willing to buy the product if they perceive the total price to be f . We also assume that f ≤ cmin ,
so that firms are not prevented from setting a zero-profit total price.
       Our main interest is in studying the Nash-equilibrium outcomes of the above game played
between firms. While we fully characterize equilibrium outcomes in our main propositions, in
discussing our results we focus on conditions for and properties of deceptive equilibria—equilibria
   3
     Our assumption on unshrouding—that a single firm can educate all consumers about all additional prices at no
cost—is in the context of most real settings unrealistically extreme, especially if (as in our credit-card application)
incurring the additional prices depends partly on the consumer’s own behavior. This extreme case is theoretically
useful for demonstrating that education often does not happen even if it is very easy. We show in our working
paper (Heidhues, Kőszegi and Murooka 2012b) that if unshrouding is somewhat costly or reaches only a fraction of
consumers, a deceptive equilibrium is more likely to occur, but our qualitative results do not change.


                                                          5
in which all firms shroud additional prices with probability one. Because no firm has an incentive
to shroud if at least one firm unshrouds, there is always an unshrouded-prices equilibrium. When a
deceptive equilibrium exists, however, it is more plausible than the unshrouded-prices equilibrium
for a number of reasons. Most importantly, we show in Appendix A that whenever a deceptive
equilibrium exists in our model with no unshrouding cost, it is the unique equilibrium in the
variant of our model in which unshrouding carries a positive cost, no matter how small the cost is.
In addition, a positive-profit deceptive equilibrium is preferred by all firms to an unshrouded-prices
equilibrium. Finally, for the lowest-priced firms, the strategy they play in an unshrouded-prices
equilibrium is weakly dominated by the strategy they play in a positive-profit deceptive equilibrium.
       To simplify our statements, we define the Bertrand outcome as the outcome of a Bertrand price-
competition game with rational consumers and transparent prices: consumers buy if and only if
v − u > cmin and pay a total price equal to cmin whenever they buy, and firms earn zero profits.


2.2      Applications

2.2.1      Credit Cards

Our first economic application is credit cards. We build on a variant of Heidhues and Kőszegi’s
(2010) model of a credit market with naive time-inconsistent borrowers.4 Firms and consumers
interact over four periods, t = 0, 1, 2, 3. In period t = 0, 1, 2, a consumer aims to maximize total
utility ut /β + 3τ =t+1 uτ , where uτ is instantaneous utility in period τ . The short-term discount
               P

factor β is distributed in the population according to the cumulative distribution function Q on
[0, 1], where a ≡ maxβ∈(0,1] Q(β)(1/β − 1) exists.5 We posit that the consumer’s period-0 total
utility is relevant for welfare evaluations.
   4
      A substantial body of research in behavioral economics documents that individuals often have a taste for
immediate gratification (Laibson, Repetto and Tobacman 2007, Augenblick, Niederle and Sprenger 2013, for instance),
that such taste is associated with credit-card borrowing (Meier and Sprenger 2010), and that individuals may be
naive about their taste (Laibson et al. 2007, Skiba and Tobacman 2008). And consistent with the specific hypothesis
that consumers underestimate costly borrowing on their credit cards, Ausubel (1991) finds that consumers are less
responsive to the post-introductory interest rate in credit-card solicitations than to the teaser rate, even though the
former is more important in determining the amount of interest they will pay.
   5
      For presentational simplicity, we use a specification in which β divides current utility instead of the (behav-
iorally equivalent) more conventional specification in which β multiplies future utility. A sufficient condition for
maxβ∈(0,1] Q(β)(1/β − 1) to exist is that Q(·) is continuous and limβ→0 Q(β)/β = 0.



                                                          6
Issuer n’s cost of managing a consumer’s account is cn < a, and all issuers acquire funds at an
interest rate of zero. In period 0, each issuer chooses salient charges (such as annual fees) fn ∈ R to
be paid in period 3, and a gross interest rate Rn ≥ 1. Simultaneously with its pricing decision, each
firm decides whether to unshroud the costs associated with credit-card use. If no firm unshrouds,
consumers assume that they have β = 1. If a firm unshrouds, consumers learn that their β’s are
drawn from Q(·), but they do not learn their individual β’s.
    Observing firms’ offers, in period 0 a consumer decides whether to get a credit card, valuing its
future convenience use at v, and the outside option (such as getting a debit card) at u. A consumer
can get multiple cards, but needs only one for convenience use, and gets other card(s) only if she
sees a strict further benefit. If a consumer gets at least one credit card, she runs up charges of 1
in period 1.6 If a consumer is indifferent between multiple credit-card deals, she lexicographically
applies an exogenously given preference over cards to decide which one to get and charge.7 If she
charges card n, she decides in period 2 whether to repay 1 to firm n immediately, or repay Rn in
period 3.8 A unit of credit-card spending generates a unit of instantaneous utility, and repaying a
unit of outstanding debt costs a unit of instantaneous utility.
    We now show that the above credit-card pricing game simplifies to our reduced-form model.
To facilitate the statement, we say that outcomes in two Nash equilibria are the same if the two
equilibria differ at most in prices at which consumers do not buy.

Lemma 1. The sets of Nash-equilibrium outcomes when fn is unrestricted (fn ∈ R) and when
   6
     We assume that the consumer makes no charging decision. This makes little difference to the logic of our results.
Since a naive consumer does not anticipate that she will delay repayment, she is happy to charge her card for its
convenience benefit, and does not care about the interest rate on her card. Furthermore, an endogenous charging
decision makes unshrouding less desirable for a firm, since a consumer who is aware that she might pay interest can
avoid this by not using her card. Finally, for the same reason as below, a firm would not want to offer a negative
up-front price when shrouding occurs.
   7
     This assumption has the stark implication that if a competitor gets a consumer to sign up for a second, less
preferred card, it gets no interest payments from the consumer. This implication plays a central role in our proof
that there is a price floor. But a price floor can arise whenever consumers with multiple cards split their expenditures
across cards. All that is necessary is that the cost of serving additional consumers below the price floor is greater
than the revenue the firm generates from the charges it attracts from holders of multiple cards.
   8
     For simplicity, we assume that a consumer does not transfer balances to another credit card in period 2, which—
under the assumption that there is a market for transfers—she would prefer to do in our three-period model unless
the effort cost of doing so was high. As predicted by O’Donoghue and Rabin (2001) and is consistent with evidence
in Shui and Ausubel (2004), however, in a more realistic, long-horizon, setting naive borrowers may procrastinate for
a long time before finding or taking advantage of favorable balance-transfer opportunities.


                                                           7
fn is restricted to be non-negative (fn ≥ 0) are the same. Furthermore, if fn is restricted to be
non-negative, then defining an ≡ Q(1/Rn )(Rn − 1), the game is strategically equivalent to (i.e., for
any action profile generates the same payoffs as) our reduced-form model with f = 0.

       The key observation is that to avoid unprofitable consumers, firms act as if they were facing a
floor of zero on the up-front price. Intuitively, although setting fn < 0 rather than fn = 0 induces
more consumers to get firm n’s card, it does not prevent them from getting and using an alternative
card they prefer. Since firm n’s profits derive from consumers using rather than just getting its
card, the consumers it attracts with the negative up-front price are unprofitable.
       Given that fn ≥ 0 for all n, each consumer gets at most one card. The resulting interest
payment on the card, then, acts as an additional price in that it adds to the firm’s profits and
lowers the consumer’s utility by the same amount. Furthermore, if shrouding occurs, a consumer
expects to repay early, so she expects to pay no additional price, whereas if unshrouding occurs, she
fully understands how much she will pay in expectation. This logic also makes clear that beyond
interest payments, a late fee or any other unanticipated fee that transfers utility from consumers
to firms can also serve as an additional price.9
       Notice that in our framework, an issuer prefers to undercut competitors’ non-negative up-
front prices if it can require consumers to use only the issuer’s card in exchange, thereby generating
interest payments down the line. This pricing strategy is consistent with the observation that many
credit cards offer usage-contingent perks such as airline miles and rental-car insurance. But while
such a strategy avoids unprofitable naive consumers, it may disproportionately attract unprofitable
sophisticated consumers, so that to avoid these consumers firms may again act as if they were
facing a price floor. Formally, suppose that a fraction λ of consumers has convenience benefit
v 0 < v (e.g., because they like paying in cash more than others), and has β = 1 and hence does
not pay interest. If firm n lowers fn below v 0 , it attracts all these sophisticated consumers. If
λ(v 0 − cmin ) + (1 − λ)(v 0 + a − cmin ) ≤ 0, therefore, the price cut is unprofitable.
   9
     Suppose, for example, that each firm n can impose a fee ân ≥ 0 for minor infractions such as paying the bill
a few days late. A consumer will pay the fee with probability Q(ân ), but unless unshrouding occurs, at the time of
selecting a card she believes this probability will be zero. We suppose that Q(â) is continuous and maxâ Q(â)â exists.
Then, an = Q(ân )ân acts like an additional price, with the maximum additional price being ā = maxâ Q(â)â.




                                                            8
2.2.2      Mutual Funds

As a second important application, we discuss mutual funds. The Investment Company Act of
1940 in the US regulates the institutional structure, including the trading prices and accounting,
of mutual funds. A fund must calculate its net asset value (NAV) based on market valuations,
and including investment advisory fees and other expenses to date, at least once daily. The price
of a share is simply the NAV divided by the number of shares, and investors can buy or redeem
shares only at the per-share NAV. In addition, a fund can impose purchase fees (front loads).10
Management fees are subject to a fiduciary duty, which we interpret as putting a legal bound on
the management fee.11 While disclosure regulations are in place to make sure investors understand
the costs of fund ownership, different types of empirical evidence suggest that investors do not fully
understand the management fees, and that they appreciate front loads better.12
       Formally, we assume that funds and investors interact over three periods, t = 0, 1, 2. In period
0, funds choose up-front prices (front loads) fn and management fees ân satisfying 0 ≤ ân ≤ Â,
where the upper bound comes from fiduciary duty. Simultaneously with this choice, firms make
unshrouding decisions. Fund n’s cost is cn per consumer initially investing in the fund.
       Consumers make investment decisions in period 0, starting off with wealth v. Consumers’
  10
      See rules 2a-4 and 22c-1 adopted under the Investment Company Act, or the summaries by the Securities and
Exchange Commission available at http://www.sec.gov/answers/nav.htm and http://www.sec.gov/divisions/
investment/icvaluation.htm (accessed October 1, 2014). A fund can also impose redemption fees (rear loads), but
since these are not important for our arguments, we do not incorporate them in our model.
   11
      See Section 36 of the Investment Company Act. The Act does not impose a specific maximum fee. Nevertheless,
fiduciary duty means that a manager must act in a way that benefits investors, and a management fee that leads
an investor into (say) an almost certain loss clearly violates this restriction. Indeed, some lawsuits have successfully
challenged excessive management fees on the basis of violating fiduciary duty: http://www.pionline.com/article/
20111212/PRINT/312129971/wal-mart-merrill-lynch-settle-401k-fee-suit (accessed October 1, 2014).
   12
      Using a natural experiment in India, Anagol and Kim (2012) show that mutual-fund investors are less sensitive to
amortized “initial issue expenses” than to otherwise identical “entry loads” paid up front. In a laboratory experiment
on portfolio choice between S&P 500 index funds with different fees, Choi, Laibson and Madrian (2010) find that
subjects, including Wharton MBA students, Harvard undergraduates, and Harvard staff members, overwhelmingly fail
to minimize fees. Using administrative data from the privatized Mexican Social Security system, Duarte and Hastings
(2012) show that investors were not sensitive to fees in choosing between funds, leading to high management fees
despite apparently strong competition. Barber, Odean and Zheng (2005) find a strong negative relationship between
mutual-fund flows and front loads, but no relationship between flows and operating expenses, and Wilcox (2003)
also presents evidence that the overwhelming majority of consumers overemphasize loads relative to expense ratios.
Based on a survey of 2,000 mutual-fund investors, Alexander, Jones and Nigro (1998) report that only 18.9% could
provide any estimate of their largest mutual fund’s expenses, although 43% claim that they knew this information at
the time of purchase.



                                                           9
outside option generates utility u in period 2. We normalize the gross return a fund can achieve
to 1, so that fund n’s net return is 1 − ân , but consumers perceive the gross return to be R̃ ≥ 1.
If a consumer selects fund n, then with probability Q(ân ), she becomes dissatisfied with managed
funds in period 1 and switches to an alternative investment opportunity with a return of 1.13 We
suppose that Q(·) is continuous and increasing, and that if investing in the mutual fund is optimal
(i.e., v(1 − ân )2 ≥ u), then Q(ân ) = 0. When choosing a mutual fund, consumers are aware of the
up-front prices, but unless at least one firm unshrouds, they ignore management fees. Unshrouding
reveals the expected lifetime management fees consumers will pay, but—for simplicity—we suppose
that conditional on selecting a mutual fund in period 0, consumers do not change their period-1
switching behavior in response.14
       We now show that this game is strategically equivalent to one in which firms choose appropri-
ately defined additional prices:

Lemma 2. Defining ṽ = R̃2 v, an ≡ v [ân + (1 − Q(ân ))(1 − ân )ân ] and a ≡ maxâ∈[0,Â] v[â + (1 −
Q(â))(1 − â)â], the mutual-fund model is strategically equivalent to our reduced-form model.

       The reason is essentially the same as in our credit-card model: the management fees generate
profits to firms and lower consumers’ utility by the same amount, and consumers are aware of it if
and only if unshrouding occurs.
       Finally, we discuss the floor on the up-front price. The language of the rules interpreting the
Investment Company Act, and in particular the phrase and definition of purchase fees, strongly
suggests that these fees are intended to be non-negative, so that fn ≥ 0. Furthermore, the constraint
fn ≥ 0 follows directly from the provision of the Investment Company Act aimed at protecting
existing consumers from using their shares to attract new consumers. Section 22(a) requires funds to
“eliminat[e] or reduc[e] so far as reasonably practicable any dilution of the value of other outstanding
  13
     This assumption is one simple way to capture the notion that higher management fees generate lower performance,
and lower performance—independently of whether a consumer realizes the importance of fees or understands gross
returns—can trigger dissatisfaction, redemption, and finding better alternative investment opportunities. We can
think of the alternative investment consumers switch to as a low-cost index fund. In Section 4, we endogenize
consumers’ outside option, which they may initially choose in period 0, as a low-cost index fund as well, although
this outside option may in principle also be different.
  14
     If v(1 − ân )2 < u and unshrouding occurs, consumers select the superior outside option, so our assumption about
period-1 switching behavior is inconsequential. Otherwise, consumers find the mutual fund optimal independently of
whether unshrouding occurs, so they refrain from switching.


                                                         10
securities” in favor of new consumers. One simple interpretation of this “no-dilution” rule is that
new investors cannot be treated better than existing investors. Since both pay the management
fees, an existing investor would strictly prefer to be treated as a new investor whenever fn < 0, so
fn < 0 violates no dilution. What gives us extra confidence that this interpretation is correct is
that the regulations on pricing were adopted as a way to interpret the no-dilution section of the
Act. Indeed, consistent with a hard price floor of zero, we are unaware of any mutual fund that
offers cash back, discounts, or other financial incentives to attract new investors.


2.2.3      Other Potential Applications

Mortgages. For mortgages with changing repayment terms, such as pay-option mortgages, interest-
only mortgages, and mortgages with teaser rates, many consumers underestimate the payments
they will have to make once an initial teaser period ends.15 In this environment, we think of the
consumer’s repayment costs assuming terms do not change as the up-front price and of the increase
in payments as the additional price. The additional price is limited by consumers’ ability to walk
away from the mortgage when payments increase. As we show in an earlier version of our paper
(Heidhues, Kőszegi and Murooka 2014), a floor on the up-front price can result because consumers
would find an overly low up-front price suspicious, and—concluding that “there must be a catch”—
would not buy. But because we need to know consumers’ beliefs to identify this price floor, it is
difficult to tell from primitives where the floor is.
       Bank accounts and hotels. In these applications, consumers buy a base product—account main-
tenance in the case of bank accounts and a room in the case of hotels—and can then buy an
add-on—such as costly overdrafting in the case of bank accounts or gambling entertainment in the
  15
     For instance, the Option Adjustable-Rate Mortgage (ARM) allows borrowers to pay less than the interest for a
period, leading to an increase in the amount owed and sharp increases in monthly payments when the mortgage resets
to an amortizing payment schedule. See “Interest-Only Mortgage Payments and Payment-Option ARMs—Are They
for You?,” information booklet prepared for consumers by the Board of Governors of the Federal Reserve System,
available at http://www.federalreserve.gov/pubs/mortgage_interestonly/mortgage_interestonly.pdf. Based
on existing evidence (Cruickshank 2000, Woodward and Hall 2012, Gerardi, Goette and Meier 2009, Gurun, Matvos
and Seru 2013, Bucks and Pence 2008) it is plausible to assume that consumers have a limited understanding of
mortgages in general, and of these mortgages in particular. And some features of these mortgages, such as the
Option ARM’s one- or three-month introductory interest rate, seem to serve only the purpose of deceiving borrowers
about the product’s cost.




                                                       11
case of Las Vegas hotels—whose cost they do not appreciate at the moment of purchase.16 We can
think of the base product’s price as the up-front price and of the utility loss from unexpectedly
high add-on payments as the additional price. The additional price is limited by consumers’ ex-post
demand response to add-on prices. Banks likely face a floor on the up-front price close to zero for
the same reason as do credit cards, but it is unclear whether hotels face a binding price floor.
         Printers. In the printer market, consumers do not anticipate high cartridge prices.17 To model
this situation, we think of the printer price as the up-front price, and suppose that consumers
value a printer explicitly or implicitly assuming that cartridges are free. Then, the additional price
is consumers’ loss (and firms’ gain) from positive cartridge prices. A price floor may arise from
suspicion or other sources, but we do not know whether it is binding.


3         Profitable Deception

This section analyzes our basic model, and discusses key economic implications.


3.1        Benchmarks: Non-Binding Price Floor or Sophisticated Consumers

First, we characterize equilibrium outcomes when the floor on the up-front price is not binding.
In our setting, this means that (even adding the maximum additional price) a firm cannot make
profits if it chooses an up-front price equal to the floor.

Proposition 1 (Equilibrium with Non-Binding Price Floor). Suppose f ≤ cmin − a. For any
ψ ∈ [0, 1], there exist equilibria in which unshrouding occurs with probability ψ. If shrouding occurs
in an equilibrium, firms earn zero profits, and consumers buy the product from a most-efficient
firm, pay f = cmin − a, a = a, and get utility v − cmin . If unshrouding occurs in an equilibrium, the
Bertrand outcome obtains.
    16
     For more detailed arguments that these products can be interpreted as deceptive, see Armstrong and Vickers
(2012) and Gabaix and Laibson (2006), respectively.
  17
     For instance, Hall (1997) and the UK’s Office of Fair Trading report that a large majority of consumers buying
a printer do not know the price of a cartridge. Gabaix and Laibson (2006) invoke cartridge prices as one of their
prime examples of hidden fees.




                                                        12
Proposition 1 implies that if the price floor is not binding, there is always a deceptive equilib-
rium. Since in a deceptive equilibrium consumers do not take into account additional prices when
choosing a product, firms set the highest possible additional price, making existing consumers valu-
able. Similarly to the logic of Lal and Matutes’s (1994) loss-leader model as well as that of many
switching-cost (Farrell and Klemperer 2007) and behavioral-economics theories, firms compete ag-
gressively for these valuable consumers ex ante, and bid down the up-front price until they eliminate
net profits. In addition, since with these prices a firm cannot profitably undercut competitors, no
firm has an incentive to unshroud.
   By Proposition 1, any equilibrium outcome is identical either to that in the above deceptive
equilibrium, or to that in an unshrouded-prices equilibrium, in which unshrouding occurs with
probability one. If unshrouding occurs, consumers make purchase decisions based on the total
price, so firms effectively play a Bertrand price-competition game, leading to a Bertrand outcome.
   As a second benchmark, we note what happens when all consumers are sophisticated in that
they observe the total prices fn + an and make purchase decisions based on these prices, and also
understand that the total value of a product is v. Then, we have standard Bertrand competition
in the total price, so standard arguments imply that the Bertrand outcome obtains. For this to be
the case, our assumption that f ≤ cmin is crucial.


3.2   Naive Consumers with a Binding Price Floor

Taken together, the benchmarks above imply that for profitable deception to occur, both naive
consumers must be present and the price floor must be binding. We turn to analyzing our model
when this is the case, assuming for the rest of the section that all consumers are naive and f > cn −a
for all n. This condition means that setting an up-front price equal to the floor does not preclude
a firm from making profits on its consumers.
   We first identify sufficient conditions for a deceptive equilibrium to exist. As above, in such an
equilibrium all firms set the maximum additional price a. Then, since firms are making positive
profits and hence have an incentive to attract consumers, they bid down the up-front price to f .
With consumers being indifferent between firms, firm n gets market share sn and therefore earns a


                                                 13
profit of sn (f + a − cn ). For this to be an equilibrium, no firm should want to unshroud additional
prices. If unshrouding occurs, consumers are willing to pay exactly v − u for the product, so firm
n can make profits of at most v − u − cn by unshrouding. Hence, unshrouding is unprofitable for
firm n if the following “Shrouding Condition” holds:

                                         sn (f + a − cn ) ≥ v − u − cn .                                       (SC)

A deceptive equilibrium exists if (SC) holds for all n. Furthermore, since sn < 1, (SC) implies that
f + a > v − u, so that in a deceptive equilibrium consumers are worse off than with their outside
option. Proposition 2 summarizes this result, and characterizes equilibria fully:18

Proposition 2 (Equilibrium with Binding Price Floor). Suppose f > cn − a for all n.
    I. If (SC) holds with a strict inequality for all n, there is a deceptive equilibrium, in which all
firms offer the contract (fn , an ) = (f , a) with probability one, consumers receive utility less than
their outside option, and firms earn positive profits. In any other equilibrium, unshrouding occurs
with probability one and the Bertrand outcome obtains.
    II. If (SC) is violated for some n, unshrouding occurs with probability one and the Bertrand
outcome obtains.

    The intuition for why firms might earn positive profits despite facing Bertrand-type price com-
petition is in two parts. First, as in previous models and as in our model with a non-binding price
floor, firms make positive profits from the additional price, and to obtain these ex-post profits each
firm wants to compete for consumers by offering better up-front terms. But once firms hit the price
floor, they exhaust this form of competition without dissipating all ex-post profits.
    Second, since a firm cannot compete further on the up-front price, there is pressure for it
to compete on the additional price—but this requires unshrouding and is therefore an imperfect
substitute for competition in the up-front price. If a firm unshrouds and cuts its additional price
by a little bit, a consumer learns not only that the firm’s product is the cheapest, but also that
all products are more expensive than she thought. If the consumer receives negative surplus from
  18
     We ignore the knife-edge case in which no firm violates (SC) but (SC) holds with equality for some firm n0 . The
equilibrium outcomes characterized in Case I of Proposition 2 remain equilibrium outcomes in this case, but there
can be additional equilibrium outcomes.


                                                         14
buying at the current total price (i.e., if f + a > v − u), this surprise leads her not to buy, so the
firm can attract her by unshrouding only if it cuts the additional price by a discrete margin. Since
this may be unprofitable, the firm may prefer not to unshroud.
    As the flip side of the above logic, if purchasing at the total market price is optimal (i.e., if
f + a ≤ v − u), then despite the negative surprise a consumer is willing to buy from a firm that
unshrouds and undercuts competitors’ additional prices by a little bit, so in this case a deceptive
equilibrium does not exist. Our model therefore says that deception—and positive profits for
firms—must be associated with suboptimal consumer purchase decisions.19
    Part II of Proposition 2 establishes that (SC) is not only sufficient, but also necessary for
profitable deception to occur. To understand the rough logic, assume toward a contradiction that
all firms shroud with positive probability. Then, a firm can ensure positive profits by shrouding and
choosing prices (f , a), so that in equilibrium firms earn positive expected profits. Since a firm that
sets the highest total price when unshrouding has zero market share if some other firm unshrouds,
to earn positive profits it must be that with positive probability all rivals set higher total prices
when shrouding. Whenever a firm sets one of these high total prices, the only event in which it
can make profits is when shrouding occurs, so we can think of its incentives by conditioning on
this event. Doing so, arguments akin to those above imply that in this high range firms set prices
(f , a), and hence firm n earns sn (f + a − cn ). But firm n can earn v − u − cn by unshrouding, so a
firm that violates (SC) prefers to unshroud.
    Note that if a consumer learns about the additional price before paying any of it and can at
that point switch without incurring any cost, then a ≤ v − u must hold. This implies that if in
addition f = 0, (SC) is violated, so a deceptive equilibrium cannot occur. In our main applications,
however, these prerequisites for (SC) to fail are violated, and in particular there is no consideration
   19
       Part I of Proposition 2 also states that whenever a deceptive equilibrium exists, there is also an equilibrium in
which unshrouding occurs with probability one and—because prices are then transparent and the standard Bertrand
logic holds—the Bertrand outcome obtains. As in the case of a non-binding price floor (Proposition 1), since un-
shrouding is costless, if a firm unshrouds it is (weakly) optimal for other firms to unshroud as well. Unlike in the case
of a non-binding price floor, however, there is no equilibrium in which unshrouding occurs with an interior probability.
Our proof of this claim elaborates on the following contradiction argument. Suppose that shrouding occurs with an
interior probability, and consider a firm that unshrouds with positive probability and sets the highest price of any
firm conditional on unshrouding. When setting this highest price, the firm earns profits only if all other firms shroud,
and even then it earns at most v − u − cn . If it instead shrouds and sets (f , a), then it earns at least sn (f + an − cn )
if all other firms also shroud. By (SC), therefore, the firm prefers to shroud with probability 1, a contradiction.


                                                            15
that limits the potential size of a relative to v − u. For credit cards, the consumer accumulates
debt before she realizes her true β, so she cannot walk away without consequences. For mutual
funds, the consumer pays management fees before realizing it. More generally, in many or most
applications, going back to the pre-contracting situation is quite costly. In the next subsection, we
analyze in more detail circumstances under which profitable deception does versus does not occur.


3.3      Key Economic Implications

We first distinguish two cases according to whether the product is socially valuable or wasteful.
       Non-vanishingly socially valuable product (there is an  > 0 such that v − u > cn +  for all n).
In this case, the right-hand side of (SC) is positive and bounded away from zero. Hence, a firm with
a sufficiently low sn violates (SC)—since it earns low profits from deception, it prefers to attract
consumers through unshrouding—and thereby guarantees that only a zero-profit unshrouded-prices
equilibrium exists. This implies that a deceptive equilibrium can exist if the number of firms is
sufficiently small, but not if the number of firms is large—as some firm will then violate (SC).
Furthermore, an increase in the number of firms can induce a regime shift from a high-price,
deceptive equilibrium to a low-price, transparent equilibrium.20
       Socially wasteful product (v − u < cn for all n). In this case, the right-hand side of (SC) is
negative while the left-hand side is positive. Hence, a deceptive equilibrium exists regardless of the
industry’s concentration and other parameter values:

Corollary 1 (Wasteful Products). Suppose f > cmin − a and v − u < cn for all n. Then, a
profitable deceptive equilibrium exists.

This perverse result has a simple logic: since a socially wasteful product cannot be profitably sold
once consumers understand its total price, a firm can never profit from coming clean.
       Assuming that the outside option is an appropriately chosen alternative product, our main
applications seem to fit this case of our model. A low-cost debit card tied to the consumer’s bank
  20
      The reason for stating our result for non-vanishingly socially valuable products is that if the social value of a
firm’s product (v − u − cn ) could be arbitrarily small, then even a firm with low profits from deception might not be
willing to unshroud. A precise condition for when unshrouding must occur is N > (f + a)/. Then, sn < /(f + a)
for some n, and for this n we have sn (f + a − cn ) <  < v − u − cn , in violation of (SC).


                                                          16
account serves the same convenience benefit as a credit card, so it is a superior outside option for
many consumers. Consistent with this perspective, Laibson et al. (2007) estimate that owning a
credit card makes the average household whose head has a high-school degree but not a college
degree worse off by the equivalent of paying $2,000 at age 20. Similarly, although not everyone agrees
with this view, many researchers believe that because few mutual-fund managers can persistently
and significantly outperform the market, most managed funds are inferior to low-cost index funds.21
And while a sharply increasing mortgage-payment schedule makes sense for the few consumers who
can look forward to drastic increases in income (as argued by Cocco 2013), the vast majority in
this market would have been better served by traditional mortgages. Our model says that such
products are sold and remain profitable in a seemingly competitive market not despite, but exactly
because they are socially inferior to alternatives.
       Indeed, the pricing structure of credit cards and mutual funds is consistent with our deceptive
equilibrium. DellaVigna and Malmendier (2004) document that most of the popular credit cards
in the US offer a $0 annual fee and a high regular interest rate.22 And many mutual funds have
zero salient front loads, but substantial management fees.23
       The different logic of socially valuable and socially wasteful industries in our model yields two
potentially important further points. First, our theory implies that if an industry experiences a
lot of entry and does not come clean in its practices, it is likely to be a socially wasteful industry.
Second, our theory suggests a general competition-impairing feature in valuable industries that is
not present in wasteful industries: to reduce the motive to deviate from their preferred positive-
  21
      See, for example, Gruber (1996), Carhart (1997), Kosowski, Timmermann, Wermers and White (2006), and
French (2008). Berk and Green (2004) and Berk and van Binsbergen (2012), however, argue that the evidence is
consistent with the hypothesis that most managers can outperform the market, and that managed funds are not
inferior to index funds.
   22
      There are two main categories of exceptions to the $0 annual fee. First, charge cards—which require repayment
within 30 days—often have an annual fee. This is consistent with the prediction of our multi-product model below,
where sophisticated consumers who anticipate paying costly interest prefer to use a charge card with a higher up-front
price to a credit card with a lower up-front price. Second, some cards offered to consumers with bad credit have an
annual fee. This presumably compensates issuers for credit risk, a consideration that is outside our model.
   23
      In the US market, our model best fits level-load and no-load mutual fund shares ($6,748 billion in total net
assets in 2012 according to the ICI fact book 2013, Figure 5.11), which generally have zero front loads and hence face
a binding floor on the up-front price. Front-end load shares ($1881 billion in total net assets in 2012) have positive
front loads, and are therefore a less obvious fit for our model with a binding price floor. But the typical such fund
uses the front load largely to pay intermediaries’ commissions, so the up-front price collected by the fund is often at
zero and hence again at the binding floor.


                                                          17
profit deceptive equilibrium in a valuable industry, each firm wants to make sure competitors earn
sufficient profits from shrouding. This feature is likely to have many implications beyond the
current paper (as, for instance, for innovation incentives in Heidhues et al. 2012), and implies that
wasteful industries may sometimes be more fiercely competitive than valuable ones.
       Misprediction of value. We now turn to some implications of Proposition 2 for consumer mis-
prediction of value. An economically relevant possibility is one where consumers underestimate the
value by a significant amount.

Corollary 2 (Unanticipated Valuable Add-On). If v − ṽ ≥ a, then in any equilibrium unshrouding
occurs with probability one.

       The condition v − ṽ ≥ a holds if consumers are completely unaware of a valuable add-on that
they can purchase after getting the product, and the outside option does not have such add-ons. In
this case, a deceptive equilibrium does not exist. Intuitively, if consumers are unaware of something
they would be happy to pay for, then unshrouding cannot make them less willing to buy the product,
so unshrouding and undercutting competitors is a profitable deviation from a candidate deceptive
equilibrium. For instance, if consumers are unaware of some valuable features of smartphones,
producers have all the incentive to explain them.
       Corollary 2 has a potentially important implication regarding the interpretation of unantici-
pated borrowing in the credit-card market. Suppose that—unlike in our model with naive time-
inconsistent borrowers—unanticipated borrowing generates unanticipated value for a consumer, for
instance by facilitating an unexpected important expense.24 Then, Corollary 2 implies that un-
shrouding should take place. From the perspective of our model, therefore, the observation that
unshrouding has not taken place implies that expensive credit-card borrowing is not only unantic-
ipated, but—consistent with a time-inconsistent setup—also initially unwanted by consumers.
       Beyond time inconsistency, our applications identify two further reasons for an unanticipated
payment not to go along with unanticipated product value. First, consumers may simply fail to
  24
     Formally, we can modify our model in Section 2.2 by assuming that the consumer is time consistent, values the
convenience benefit of the card at ṽ, and can in period 2 spend 1 on something she values at 1 + v − ṽ. She does not
anticipate this need ex ante, and if she decides to spend, she cannot repay her credit-card balance. This implies that
she is willing to pay at most a = v − ṽ to delay repayment.



                                                         18
anticipate a charge—such as a late fee for credit cards or management fee for mutual funds—that
is imposed without creating any additional value. Second, consumers may understand that there
is a valuable add-on—e.g., that a printer accommodates cartridges for printing—but explicitly or
implicitly underestimate the add-on price or not think about add-on expenses when buying the
base product.
    Now suppose that consumers overestimate the product’s value (ṽ > v). Holding other parame-
ters constant, this does not affect (SC), and hence does not affect whether a deceptive equilibrium
exists in our model. Going slightly beyond our assumptions, however, consumers’ overestimation of
value can facilitate the creation of a socially wasteful industry by making consumers more willing
to buy the product. In particular, if v − f < u < ṽ − f , then a profitable deceptive equilibrium
exists when consumers perceive the value to be ṽ, but not when they perceive the value to be
v. For instance, the popularity of managed mutual funds is probably due not only to consumers’
underestimation of their expenses, but also to consumers’ overestimation of their gross returns.


4     Sophisticated Consumers and Multi-Product Markets

In this section, we incorporate sophisticated consumers—who observe but cannot avoid additional
prices—into our model, considering both single- and multi-product markets. We assume that the
proportion of sophisticated consumers is κ ∈ (0, 1), that the price floor is binding (f > cn − a for
all n), and that consumers perceive the product’s value correctly (ṽ = v).


4.1   Sophisticated Consumers in Our Basic Model

Notice that in any deceptive equilibrium, sophisticated consumers do not buy the product: if they
did, they would buy from a firm with the lowest total price, and such a firm would prefer to either
undercut equal-priced competitors and attract sophisticated consumers, or (if there are no equal-
priced competitors) to unshroud and attract all consumers. Hence, the total price of any firm
exceeds v − u. This implies that if firm n unshrouds, it maximizes profits by setting a total price
equal to v − u, attracting all naive and sophisticated consumers. Combining these considerations,



                                                19
unshrouding is unprofitable if

                                 (1 − κ)sn (f + a − cn ) ≥ v − u − cn ,                              (1)

and a deceptive equilibrium exists if and only if Condition (1) holds for all n.
   If the product is socially valuable, the condition for a deceptive equilibrium to exist is stricter if
sophisticated consumers are present than if they are not. Intuitively, while sophisticated consumers
do not buy the product when the additional price is high, they can be attracted by a price cut,
creating pressure to cut the additional price—and by implication also to unshroud. This result
is consistent with the insights of Gabaix and Laibson (2006) and Armstrong and Vickers (2012)
that if the proportion of sophisticated consumers is sufficiently high, transparency and efficiency
obtains. But if the product is socially wasteful, then as before a profitable deceptive equilibrium
always exists. The reason is simple: sophisticated consumers never buy a socially wasteful product
in equilibrium, so their presence is irrelevant—firms just exploit naive consumers.


4.2   Sophisticated Consumers with an Alternative Transparent Product

In many markets for deceptive products, alternatives that are more transparent than and arguably
superior to the deceptive products exist. Motivated by this observation, we modify our model
above by assuming that each firm has a transparent product in addition to the potentially deceptive
product we have described. Because we think of the alternative product as an endogenous outside
option, we set consumers’ utility from not buying, u, to zero. Consumers’ value for the transparent
product is w > 0, and firm n’s cost of producing it is cw
                                                        n ≥ 0, with at least two firms having the

lowest cost cw            w                                                                 w
             min = minn {cn }. Crucially, we posit that product w is socially valuable (w −cmin > 0),

and is not inferior to product v: w − cw
                                       min ≥ v − cmin . Consumers are interested in buying at most

one product. Firms simultaneously set the up-front and additional prices for product v, the single
transparent price for product w, and decide whether to unshroud the additional prices of product
v. If consumers weakly prefer buying and are indifferent between a number of firms in the market
for product v or w, firms split the respective market in proportion to sn or sw
                                                                              n , respectively; and

if consumers are indifferent between products v and w, a given positive fraction of them chooses


                                                  20
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